Q. 9 聽Give a smooth parametrization,... [FREE SOLUTION]

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Chapter 14: Q. 9 (page 1119)

Give a smooth parametrization, in terms of u and v, of the sphere of radius k and centered at the origin.

Short Answer

Expert verified

$T he required parametrization is r (u, v) = <ksinv cosu, ksinv sinu, k \cos v> 0 鈮� u 鈮� 2 蟺 and 0 鈮� v 鈮� 蟺$

Step by step solution

Step 1. Given

The given equation of the sphere of radius r and centered at the origin is

$x^{2} + y^{2} + z^{2} = k^{2} . . . . . . . . . (1)$

Step 2. Parametrization of sphere

$L e t x^{2} + y^{2} = a^{2} be the parametrization of the circle \{\begin{cases} x = a \cos u \\ y = a \sin u \end{cases} where, 0 鈮� u 鈮� 2 蟺 Then (1) becomes x^{2} + y^{2} + z^{2} = k^{2} a^{2} + z^{2} = k^{2} The parametrization of the circle a^{2} + z^{2} = k^{2} is \{\begin{cases} a = ksinv \\ z = k \cos v \end{cases} The parametrization of the sphere x^{2} + y^{2} + z^{2} = k^{2}, is \{\begin{cases} x = a cosu = ksinv cosu \\ y = a sinu = ksinv sinu \\ z = k \cos v \end{cases} where, 0 鈮� u 鈮� 2 蟺 and 0 鈮� v 鈮� 蟺 Therefore the required parametrization is r (u, v) = <ksinv cosu, ksinv sinu, k \cos v> 0 鈮� u 鈮� 2 蟺 and 0 鈮� v 鈮� 蟺$

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91影视

Give a smooth parametrization, in terms of u and v, of the sphere of radius k and centered at the origin.

Short Answer

Step by step solution

Step 1. Given

Step 2. Parametrization of sphere

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